Critical Exponent for the Density of Percolating Flux

TL;DR

This paper calculates the critical exponent for the density of a percolating flux network in a model related to gauge theory and the 3D Ising model, revealing rapid flux redistribution near the phase transition.

Contribution

It provides a novel calculation of the critical exponent for the percolating flux cluster density using the epsilon-expansion, linking flux properties to phase transition behavior.

Findings

The percolating cluster density exponent is ζ = (1−α) − (φ−1).

Flux redistributes from infinite to finite clusters faster than total density decreases.

The critical exponents α and φ are computed via epsilon-expansion.

Abstract

This paper is a study of some of the critical properties of a simple model for flux. The model is motivated by gauge theory and is equivalent to the Ising model in three dimensions. The phase with condensed flux is studied. This is the ordered phase of the Ising model and the high temperature, deconfined phase of the gauge theory. The flux picture will be used in this phase. Near the transition, the density is low enough so that flux variables remain useful. There is a finite density of finite flux clusters on both sides of the phase transition. In the deconfined phase, there is also an infinite, percolating network of flux with a density that vanishes as $T \rightarrow T_{c}^{+}$. On both sides of the critical point, the nonanalyticity in the total flux density is characterized by the exponent $(1-\alpha)$. The main result of this paper is a calculation of the critical exponent for the percolating network. The exponent for the density of the percolating cluster is $ \zeta = (1-\alpha) - (\varphi-1)$. The specific heat exponent $\alpha$ and the crossover exponent $\varphi$ can be computed in the $\epsilon$-expansion. Since $\zeta < (1-\alpha)$, the variation in the separate densities is much more rapid than that of the total. Flux is moving from the infinite cluster to the finite clusters much more rapidly than the total density is decreasing.